Since giving the Einstein tensor does not fully determine the Riemann tensor, but leaves the Weyl tensor unspecified ( see the Ricci decomposition ), the Einstein equation may be considered a kind of compatibility condition: the spacetime geometry must be consistent with the amount and motion of any matter or nongravitational fields, in the sense that the immediate presence " here and now " of nongravitational energy-momentum causes a proportional amount of Ricci curvature " here and now ".

However, the more difficult proof of the converse ( see below ) makes use of the vector space structure: Since both of the factors are vector spaces over Q, the tensor product can be taken over Q.

Since χ is a tensor, P is not necessarily colinear with E.

Since the trace of any tensor is independent of any coordinate system, the most complete coordinate-free decomposition of a symmetric tensor is to represent it as the sum of a constant tensor and a traceless symmetric tensor.

Since the tensor product concept is independent of any choice of basis, taking the tensor product of two vector bundles on M is routine.

Since A and B may both be regarded as R-modules, we may form their tensor product

Since the covariant derivative of a tensor field at a point depends only on value of the vector field at one can define the covariant derivative along a smooth curve in a manifold:

Since the dual of the pseudoscalar is the product of two " pseudo-quantities " it can be seen that the resulting tensor is a true tensor, and does not change sign upon an inversion of axes.

( Since cars have reasonable symmetrical shapes, the off-diagonal terms of the angular inertia tensor can usually be ignored.

Since the Riemann curvature depends only on the Levi-Civita connection, the variation of the Riemann tensor can be calculated as,

Since electromagnetic fields contribute only a traceless term to the stress-energy tensor,

Since the mixed Kronecker delta is equivalent to the mixed metric tensor,

Since Hilbert spaces have inner products, one would like to introduce an inner product, and therefore a topology, on the tensor product that arise naturally from those of the factors.

Since D' and N' commute with one another and Af, we see that D' and N' commute with T.

Since there is no evidence that late Maastrichtian nonavian dinosaurs could burrow, swim or dive, they were unable to shelter themselves from the worst parts of any environmental stress that occurred at the K – T boundary.

Since the map d restricts to 0 on I < sub > x </ sub >< sup > 2 </ sup > ( the reader should verify this ), d descends to a map from I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup > to the dual of the tangent space, ( T < sub > x </ sub > M )< sup >*</ sup >.

Since tension is defined as the force that the chain exerts on itself, T must be parallel to the chain.

Since then it has been open source software, originally under a license particular to AT & T but, since the 93q release in early 2005, it has been licensed under the Common Public License.

Since the column vectors of A are the row vectors of the transpose of A ( denoted here by A < sup > T </ sup >), column rank of A equals row rank of A is equivalent to saying that the rank of a matrix is equal to the rank of its transpose, i. e. rk ( A ) = rk ( A < sup > T </ sup >).

Since this construction works for any vector space V, we conclude that T is a functor from K-Vect to K-Alg.

Since the theorem was also discovered independently by E. T. Whittaker, by Vladimir Kotelnikov, and by others, it's also known as the Nyquist – Shannon – Kotelnikov, Whittaker – Shannon – Kotelnikov, Whittaker – Nyquist – Kotelnikov – Shannon, WKS, as well as the Cardinal Theorem of Interpolation Theory.

Since social cognitive theory suggests that meaningful sources of identity can be discovered by teens who feel connected to what they ’ re viewing, reality T. V.

Since Pupin's patent contained a ( less accurate ) formula, AT & T was open to claims of incomplete disclosure.

Since T is uncountable, this subset of R must be uncountable.

Since a Hausdorff space is the same as a preregular T < sub > 0 </ sub > space, a regular space that is also T < sub > 0 </ sub > must be Hausdorff ( and thus T < sub > 3 </ sub >).

Since G states only that it is not provable in T, no contradiction is presented by its provability in T ’.

Since the potential V < sub > T </ sub > changes only in one direction, i. e. the direction toward the main body, the satellite can be expected to take an axially symmetric form.

Since its inception, 12 researchers associated with Berkeley Lab ( Ernest Lawrence, Glenn T. Seaborg, Edwin M. McMillan, Owen Chamberlain, Emilio G. Segrè, Donald A. Glaser, Melvin Calvin, Luis W. Alvarez, Yuan T. Lee, Steven Chu, George F. Smoot and Saul Perlmutter ) have been awarded the Nobel Prize.

Since the 2000 presidential election, Li Ao has bitterly spoken out against pro-independence Nobel laureate Yuan T. Lee, who publicly supported Chen Shui-bian.

Since he first came to prominence in the mid-1960s, critics have cited Campbell as one of the leading writers in his field: T. E. D. Klein has written that " Campbell reigns supreme in the field today ", while S. T. Joshi stated, " future generations will regard him as the leading horror writer of our generation, every bit the equal of Lovecraft or Blackwood.

Since 2003, the team has been forced on an extended road trip for much of February since the AT & T Center hosts the San Antonio Stock Show & Rodeo.

Since n grows faster than log < sub > 2 </ sub >( n ), there must exist a value n < sub > 0 </ sub > such that

This subextension is called a separable closure of K. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of K < sup > sep </ sup >, of degree > 1.

Since the 1990s, SO < sub > 2 </ sub > emissions have dropped 40 %, and according to the Pacific Research Institute, acid rain levels have dropped 65 % since 1976.

Since B < sub > n </ sub >= 0 for all odd n > 1, and many formulas only involve even-index Bernoulli numbers, some authors write B < sub > n </ sub > instead of B < sub > 2n </ sub >.

Since the mass numbers of these are 12, 13 and 14 respectively, the three isotopes of carbon are known as carbon-12, carbon-13, and carbon-14, often abbreviated to < sup > 12 </ sup > C, < sup > 13 </ sup > C, and < sup > 14 </ sup > C.

Since at this stress the slope of the material's stress-strain curve, E < sub > t </ sub > ( called the tangent modulus ), is smaller than that below the proportional limit, the critical load at inelastic buckling is reduced.

Since we are assuming circular motion, let r ( t ) = R · u < sub > r </ sub >, where R is a constant ( the radius of the circle ) and u < sub > r </ sub > is the unit vector pointing from the origin to the point mass.

Since < tt > ns1 </ tt > is contained in < tt > example. org </ tt >, this requires resolving < tt > example. org </ tt > first, which presents a circular dependency.

Since every equivalence relation over X corresponds to a partition of X, and vice versa, the number of possible equivalence relations on X equals the number of distinct partitions of X, which is the nth Bell number B < sub > n </ sub >:

Since all probabilities p < sub > i </ sub > add up to one: p < sub > 1 </ sub > + p < sub > 2 </ sub > + ... + p < sub > k </ sub > = 1, the expected value can be viewed as the weighted average, with p < sub > i </ sub >’ s being the weights: